Problems

\(A, B, C, D\) are four collinear points whose cross-ratio \((ABCD)\) is \(-\tan^2\theta\). Find, in terms of \(\theta\), the cross-ratios \((ADCB), (ACBD), (ADBC), (ABDC), (ACDB)\). \par Four points \(P_1, P_2, P_3, P_4\), collinear with \(A, B, C\), are such that \((ABCP_r) = -\tan^2\theta_r\), \(r=1,2,3,4\). Prove that \[ (P_1 P_2 P_3 P_4) = \frac{\sin(\theta_2-\theta_1)\sin(\theta_4-\theta_3)}{\sin(\theta_4-\theta_1)\sin(\theta_2-\theta_3)} \frac{\sin(\theta_2+\theta_1)\sin(\theta_4+\theta_3)}{\sin(\theta_4+\theta_1)\sin(\theta_2+\theta_3)}. \]

If the sides of two triangles touch a conic, prove that their vertices all lie on a conic. \par State the relation between the foci of a conic and the circular points and deduce that, if a parabola touches each of three given straight lines, then its focus lies on the circumcircle of the triangle formed by these lines.

Prove that the circles described on the three diagonals of a complete quadrilateral are coaxal.

Express \[ \begin{vmatrix} \sin^3\theta & \sin\theta & \cos\theta \\ \sin^3\alpha & \sin\alpha & \cos\alpha \\ \sin^3\beta & \sin\beta & \cos\beta \end{vmatrix} \] as the product of four sines and hence find all values of \(\theta\), in terms of \(\alpha\) and \(\beta\), for which the value of this determinant is zero.

If \(O, H, I, K\) are respectively the centres of the circum-, ortho-, in-, and nine-point-circles of a triangle \(ABC\), prove that \(IH^2 = 2r^2-4R^2\cos A\cos B\cos C\), and obtain similar expressions for \(OI^2\) and \(OH^2\), where \(r, R\) are the radii of the in- and circum-circles. \par Deduce that \(IK = \frac{1}{2}R-r\), and hence that the in- and nine-point-circles touch.

\(P\) is any point on the circumcircle of a triangle \(ABC\) and \(A', B', C'\) are the other ends of the diameters through \(A, B, C\) respectively; if \(PA', PB', PC'\) meet \(BC, CA, AB\) respectively in \(X, Y, Z\), prove, by using Pascal's Theorem or otherwise, that \(X, Y, Z\) lie on the same diameter of the circumcircle.

The lines joining the vertices \(A, B, C\) of a triangle to a point \(P\) cut the opposite sides in \(L, M, N\) respectively; if \(X, Y, Z\) are the middle points of \(MN, NL, LM\) respectively, prove that the lines \(AX, BY, CZ\) are concurrent.

Prove that the foot of the perpendicular from the focus of a parabola to a variable tangent lies on a fixed line. \par If \(S\) is a focus of an ellipse, prove that the parabola with its focus at \(S\) touching the tangent and normal at a point of the ellipse also touches the minor axis of the ellipse.

\(BC, AD\) are two chords of a conic through a focus \(P\) of the conic; if \(CA, BD\) meet at \(Q\) and \(AB, CD\) meet at \(R\), prove that \(QR\) subtends a right angle at the focus \(P\).

Prove that the inverse of a circle with respect to a sphere is in general another circle, but may specially be a straight line. \par The lines through a fixed point \(O\) and the points of a fixed circle \(\Gamma\) generate a cone; prove that any plane parallel to the plane of \(\Gamma\) cuts the cone in a circle and that there is another set of parallel planes cutting the cone in circles. \par Find the relation between \(O\) and \(\Gamma\), if the two sets of parallel planes are the same.